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Holden Lyu
Holden Lyu
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Uniqueness of Cauchy problem of constant coefficient differential operator with given boundary information on compact domain

I'm considering the cauchyuniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have

\begin{equation}\label{overdetermined problem of constant coefficient} \left\{\begin{aligned} Au&=0&\hspace{5pt}&\text{in}&\hspace{5pt}&\Omega\subset\mathbb{R}^n\\ \partial_\nu^ku&=f_k&\hspace{5pt}&\text{on}&\hspace{5pt}&\partial\Omega \end{aligned}\right. \end{equation}

Whene can we have uniqueness and existence of solutions? For example, I know when $A=\Delta$ and $k=0$ there exists one unique solution. I'm wondering if we can have some similar results for any constant coefficient differential operator.

By the way, I'm not sure whether I need to substitude 'for' by 'of' in the phrase 'the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$' or not. Please let me know if I were wrong.

Uniqueness of Cauchy problem of constant coefficient differential operator on compact domain

I'm considering the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ such that we have

\begin{equation}\label{overdetermined problem of constant coefficient} \left\{\begin{aligned} Au&=0&\hspace{5pt}&\text{in}&\hspace{5pt}&\Omega\subset\mathbb{R}^n\\ \partial_\nu^ku&=f_k&\hspace{5pt}&\text{on}&\hspace{5pt}&\partial\Omega \end{aligned}\right. \end{equation}

Whene can we have uniqueness and existence of solutions? For example, I know when $A=\Delta$ and $k=0$ there exists one unique solution. I'm wondering if we can have some similar results for any constant coefficient differential operator.

By the way, I'm not sure whether I need to substitude 'for' by 'of' in the phrase 'the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$' or not. Please let me know if I were wrong.

Uniqueness problem of constant coefficient differential operator with given boundary information on compact domain

I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have

\begin{equation}\label{overdetermined problem of constant coefficient} \left\{\begin{aligned} Au&=0&\hspace{5pt}&\text{in}&\hspace{5pt}&\Omega\subset\mathbb{R}^n\\ \partial_\nu^ku&=f_k&\hspace{5pt}&\text{on}&\hspace{5pt}&\partial\Omega \end{aligned}\right. \end{equation}

Whene can we have uniqueness and existence of solutions? For example, I know when $A=\Delta$ and $k=0$ there exists one unique solution. I'm wondering if we can have some similar results for any constant coefficient differential operator.

By the way, I'm not sure whether I need to substitude 'for' by 'of' in the phrase 'the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$' or not. Please let me know if I were wrong.

Source Link
Holden Lyu
Holden Lyu
  • 431
  • 2
  • 9

Uniqueness of Cauchy problem of constant coefficient differential operator on compact domain

I'm considering the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ such that we have

\begin{equation}\label{overdetermined problem of constant coefficient} \left\{\begin{aligned} Au&=0&\hspace{5pt}&\text{in}&\hspace{5pt}&\Omega\subset\mathbb{R}^n\\ \partial_\nu^ku&=f_k&\hspace{5pt}&\text{on}&\hspace{5pt}&\partial\Omega \end{aligned}\right. \end{equation}

Whene can we have uniqueness and existence of solutions? For example, I know when $A=\Delta$ and $k=0$ there exists one unique solution. I'm wondering if we can have some similar results for any constant coefficient differential operator.

By the way, I'm not sure whether I need to substitude 'for' by 'of' in the phrase 'the cauchy problem for a constant coefficient differential operator $A$ on compact domain $\Omega$' or not. Please let me know if I were wrong.